Theorem efrirr 5665

Description: A well-founded class does not belong to itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
efrirr	⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		frirr 5661	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 E 𝐴)
2		epelg 5585	. . . 4 ⊢ (𝐴 ∈ 𝐴 → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
3	2	adantl 481	. . 3 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → (𝐴 E 𝐴 ↔ 𝐴 ∈ 𝐴))
4	1, 3	mtbid 324	. 2 ⊢ (( E Fr 𝐴 ∧ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴)
5	4	pm2.01da 799	1 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   class class class wbr 5143   E cep 5583   Fr wfr 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637

This theorem is referenced by:  tz7.2  5668  ordirr  6402